ALBERT-L SZL BARAB SI NETWORK SCIENCE THE BARAB SI-ALBERT MODEL ACKNOWLEDGEMENTS M RTON P SFAI

ALBERT-LÁSZLÓ BARABÁSI
NETWORK SCIENCE
MÁRTON PÓSFAI
GABRIELE MUSELLA
MAURO MARTINO
ROBERTA SINATRA
ACKNOWLEDGEMENTS SARAH MORRISON
AMAL HUSSEINI
PHILIPP HOEVEL
THE BARABÁSI-ALBERT MODEL
5

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
INDEX
Figure 5.0 (cover image)
Scale-free Sonata
Composed by Michael Edward Edgerton in 2003,
1 sonata for piano incorporates growth and pref-
erential attachment to mimic the emergence of
a scale-free network. The image shows the begin-
ning of what Edgerton calls Hub #5. The relation-
ship between the music and networks is explained
by the composer:
“6 hubs of different length and procedure were
distributed over the 2nd
and 3rd
movements. Mu-
sically, the notion of an airport was utilized by
diverting all traffic into a limited landing space,
while the density of procedure and duration were
varied considerably between the 6 differing occur-
rences.“
Introduction
Growth and Preferential Attachment
The Barabási-Albert Model
Degree Dynamics
Degree Distribution
The Absence of Growth or Preferential Attachment
Measuring Preferential Attachment
Non-linear Preferential Attachment
The Origins of Preferential Attachment
Diameter and Clustering Coefficient
Homework
Summary
ADVANCED TOPICS 5.A
Deriving the Degree Distribution
ADVANCED TOPICS 5.B
Nonlinear Preferential Attachment
ADVANCED TOPICS 5.C
The Clustering Coefficient
Bibliography
This book is licensed under a
Creative Commons: CC BY-NC-SA 2.0.
PDF V48 19.09.2014

3
SECTION 5.1
Hubs represent the most striking difference between a random and
a scale-free network. On the World Wide Web, they are websites with
an exceptional number of links, like google.com or facebook.com; in the
metabolic network they are molecules like ATP or ADP, energy carriers in-
volved in an exceptional number of chemical reactions. The very existence
of these hubs and the related scale-free topology raises two fundamental
questions:
• Why do so different systems as the WWW or the cell converge
to a similar scale-free architecture?
• Why does the random network model of Erdős and Rényi fail to
reproduce the hubs and the power laws observed in real
networks?
The first question is particularly puzzling given the fundamental dif-
ferences in the nature, origin, and scope of the systems that display the
scale-free property:
• The nodes of the cellular network are metabolites or proteins,
while the nodes of the WWW are documents, representing
information without a physical manifestation.
•The links within the cell are chemical reactions and binding interac-
tions, while the links of the WWW are URLs, or small segments of com-
puter code.
• The history of these two systems could not be more different: The
cellular network is shaped by 4 billion years of evolution, while
the WWW is less than three decades old.
•The purpose of the metabolic network is to produce the
chemical components the cell needs to stay alive, while the purpose of
the WWW is information access and delivery.
INTRODUCTION
>
Online Resource 5.1
Scale-free Sonata
Listen to a recording of Michael Edward
Edgerton's 1 sonata for piano, music in-
spired by scale-free networks.
>

INTRODUCTION
4
To understand why so different systems converge to a similar architec-
ture we need to first understand the mechanism responsible for the emer-
gence of the scale-free property. This is the main topic of this chapter.
Given the diversity of the systems that display the scale-free property, the
explanation must be simple and fundamental. The answers will change the
way we model networks, forcing us to move from describing a network’s
topology to modeling the evolution of a complex system.

THE BARABÁSI-ALBERT MODEL 5
GROWTH AND PREFERENTIAL
ATTACHMENT
SECTION 5.2
We start our journey by asking: Why are hubs and power laws absent in
random networks? The answer emerged in 1999, highlighting two hidden
assumptions of the Erdős-Rényi model, that are violated in real networks
[1]. Next we discuss these assumptions separately.
Networks Expand Through the Addition of New Nodes
The random network model assumes that we have a fixed number of
nodes, N. Yet, in real networks the number of nodes continually grows
thanks to the addition of new nodes.
Consider a few examples:
• In 1991 the WWW had a single node, the first webpage build by Tim
Berners-Lee, the creator of the Web. Today the Web has over a trillion
(1012
) documents, an extraordinary number that was reached through
the continuous addition of new documents by millions of individuals
and institutions (Figure 5.1a).
• The collaboration and the citation network continually expands
through the publication of new research papers (Figure 5.1b).
• The actor network continues to expand through the release of new
movies (Figure 5.1c).
• The protein interaction network may appear to be static, as we inherit
our genes (and hence our proteins) from our parents. Yet, it is not: The
number of genes grew from a few to the over 20,000 genes present in
a human cell over four billion years.
Consequently, if we wish to model these networks, we cannot resort to
a static model. Our modeling approach must instead acknowledge that
networks are the product of a steady growth process.

6
Nodes Prefer to Link to the More Connected Nodes
The random network model assumes that we randomly choose the in-
teraction partners of a node. Yet, most real networks new nodes prefer
to link to the more connected nodes, a process called preferential attach-
ment (Figure 5.2).
Consider a few examples:
• We are familiar with only a tiny fraction of the trillion or more docu-
ments available on the WWW. The nodes we know are not entirely ran-
dom: We all heard about Google and Facebook, but we rarely encoun-
ter the billions of less-prominent nodes that populate the Web. As our
knowledge is biased towards the more popular Web documents, we
are more likely to link to a high-degree node than to a node with only
few links.
• No scientist can attempt to read the more than a million scientific pa-
pers published each year. Yet, the more cited is a paper, the more likely
that we hear about it and eventually read it. As we cite what we read,
our citations are biased towards the more cited publications, repre-
senting the high-degree nodes of the citation network.
• The more movies an actor has played in, the more familiar is a casting
director with her skills. Hence, the higher the degree of an actor in the
actor network, the higher are the chances that she will be considered
for a new role.
In summary, the random network model differs from real networks in
two important characteristics:
(A) Growth
Real networks are the result of a growth process that continuously
increases N. In contrast the random network model assumes that the
number of nodes, N, is fixed.
(B) Preferential Attachment
In real networks new nodes tend to link to the more connected nodes.
In contrast nodes in random networks randomly choose their inter-
action partners.
There are many other differences between real and random networks,
some of which will be discussed in the coming chapters. Yet, as we show
next, these two, growth and preferential attachment, play a particularly im-
portant role in shaping a network’s degree distribution.
Networks are not static, but grow via the
addition of new nodes:
(a) The evolution of the number of WWW
hosts, documenting the Web’s rapid
growth. After http://www.isc.org/solu-
tions/survey/history.
(b) The number of scientific papers published
in Physical Review since the journal’s
founding. The increasing number of pa-
pers drives the growth of both the science
collaboration network as well as of the cita-
tion network shown in the figure.
(c) Number of movies listed in IMDB.com,
driving the growth of the actor network.
Figure 5.1
The Growth of Networks
(a)
(b)
(c)
WORLD WIDE WEB
ACTOR NETWORK
CITATION NETWORK
GROWTH AND PREFERENTIAL ATTACHMENT
YEARS
1880
50000
100000
150000
200000
250000
0
1900 1920 1940 1960 1980 2000 2020
NUMBER
OF
MOVIES
YEARS
YEARS
1880 1900 1920 1940 1960 1980 2000 2020
1982
0x100
1x108
2x108
3x108
4x108
5x108
6x108
8x108
9x108
1x109
7x108
1987 1992 1997 2002 2007 2012
NUMBER
OF
HOSTS
0
50000
100000
150000
200000
250000
300000
400000
450000
350000
NUMBER
OF
PAPERS

1925
XXI
1960
PUBLICATION
DATE
MILESTONES
1968 1970 1976 1980 1985 1999
1950
1945
1941
1923 1931 1935 1955 2005 2010
2000
György Pólya
1990 1995
PÓLYA PROCESS
MATHEMATICIAN
George Udmy Yule
YULE PROCESS
STATISTICIAN
György Pólya (1887-1985)
Preferential attachment made its
first appearance in 1923 in the
celebrated urn model of the
Hungarian mathematician György
Pólya [2]. Hence, in mathematics
preferential attachment is often
called a Pólya process.
Derek de Solla Price (1922-1983)
used preferential attachment to
explain the citation statistics of
scientific publications, calling it
cumulative advantage [7].
Robert Merton (1910-2003)
In sociology preferential attachment
is often called the Matthew effect,
named by Merton [8] after a passage
in the Gospel of Matthew.
Barabási (1967) & Albert (1972)
introduce the term preferential
attachment to explain the origin of
scale-free networks [1].
Herbert Alexander Simon (1916-2001)
explain the fat-tailed nature of the
distributions describing city sizes,
word frequencies, or the number of
papers published by scientists [6].
George Udmy Yule (1871-1951)
explain the power-law distribution of
the number of species per genus of
flowering plants [3]. Hence, in
statistics preferential attachment is
often called a Yule process.
Robert Gibrat (1904-1980)
proposed that the size and the
growth rate of a firm are indepen-
dent. Hence, larger firms grow
faster [4]. Called proportional growth,
this is a form of preferential
attachment.
George Kinsley Zipf (1902-1950)
explain the fat tailed distribution of
wealth in the society [5].
Robert Gibrat
PROPORTIONAL GROWTH
ECONOMIST
Albert-László Barabási & Réka Albert
PREFERENTIAL ATTACHMENT
NETWORK SCIENTISTS
Robert Merton
MATTHEW EFFECT
SOCIOLOGIST
Herbert Alexander Simon
MASTER EQUATION
POLITICAL SCIENTIST
Derek de Solla Price
CUMULATIVE ADVANTAGE
PHYSICIST
George Kinsley Zipf
WEALTH DISTRIBUTION
ECONOMIST
“For everyone who has will be
given more, and he will have
an abundance.”
Gospel of Matthew
THE
BARABÁSI-ALBERT
MODEL
GROWTH
AND
PREFERENTIAL
ATTACHMENT
FIG 5.2
PREFERENTIAL ATTACHMENT: A BRIEF HISTORY
7
Preferential attachment has emerged independently in many disciplines, helping explain
the presence of power laws characterising various systems. In the context of networks pref-
erential attachment was introduced in 1999 to explain the scale-free property.

THE BARABÁSI-ALBERT
MODEL
SECTION 5.2
The recognition that growth and preferential attachment coexist in real
networks has inspired a minimal model called the Barabási-Albert model,
which can generate scale-free networks [1]. Also known as the BA model or
the scale-free model, it is defined as follows:
We start with m0
nodes, the links between which are chosen arbitrarily,
as long as each node has at least one link. The network develops following
two steps (Figure 5.3):
(A) Growth
At each timestep we add a new node with m (≤ m0
) links that connect
the new node to m nodes already in the network.
(B) Preferential attachment
The probability Π(k) that a link of the new node connects to node i
depends on the degree ki
as
Preferential attachment is a probabilistic mechanism: A new node is
free to connect to any node in the network, whether it is a hub or has a
single link. Equation (5.1) implies, however, that if a new node has a choice
between a degree-two and a degree-four node, it is twice as likely that it
connects to the degree-four node.
After t timesteps the Barabási-Albert model generates a network with N
= t + m0
nodes and m0
+ mt links. As Figure 5.4 shows, the obtained network
has a power-law degree distribution with degree exponent γ=3. A mathe-
matically self-consistent definition of the model is provided in BOX 5.1.
As Figure 5.3 and Online Resource 5.2 indicate, while most nodes in the
network have only a few links, a few gradually turn into hubs. These hubs
are the result of a rich-gets-richer phenomenon: Due to preferential attach-
Figure 5.3
Evolution of the Barabási-Albert Model
The sequence of images shows nine subse-
quent steps of the Barabási-Albert model.
Empty circles mark the newly added node to
the network, which decides where to connect
its two links (m=2) using preferential attach-
ment (5.1). After [9].
(5.1)
k
k
k
( ) .
i
i
j
j
∑
Π =
Online Resource 5.2
Emergence of a Scale-free Network
Watch a video that shows the growth of a
scale-free network and the emergence of the
hubs in the Barabási-Albert model. Courtesy
of Dashun Wang.
>
>

The degree distribution of a network gen-
erated by the Barabási-Albert model. The
figure shows pk
for a single network of size
N=100,000 and m=3. It shows both the linear-
ly-binned (purple) and the log-binned version
(green) of pk
. The straight line is added to guide
the eye and has slope γ=3, corresponding to
the network’s predicted degree exponent.
Figure 5.4
The Degree Distribution
ment new nodes are more likely to connect to the more connected nodes
than to the smaller nodes. Hence, the larger nodes will acquire links at the
expense of the smaller nodes, eventually becoming hubs.
In summary, the Barabási-Albert model indicates that two simple
mechanisms, growth and preferential attachment, are responsible for the
emergence of scale-free networks. The origin of the power law and the as-
sociated hubs is a rich-gets-richer phenomenon induced by the coexistence
of these two ingredients. To understand the model’s behavior and to quan-
tify the emergence of the scale-free property, we need to become familiar
with the model’s mathematical properties, which is the subject of the next
section.
9
THE BARABÁSI-ALBERT MODEL THE BARABÁSI-ALBERT MODEL
pk
k
100
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
101
102
γ=3
103

b
(4.1)
INTRODUCTION
10
a
BOX 5.1
THE MATHEMATICAL DEFINITION OF THE BARABÁSI-ALBERT MODEL
The definition of the Barabási-Albert model leaves many mathe-
matical details open:
• It does not specify the precise initial configuration of the first
m0
nodes.
• It does not specify whether the m links assigned to a new node
are added one by one, or simultaneously. This leads to potential
mathematical conflicts: If the links are truly independent, they
could connect to the same node i, resulting in multi-links.
Bollobás and collaborators [10] proposed the Linearized Chord Di-
agram (LCD) to resolve these problems, making the model more
amenable to mathematical approaches.
According to the LCD, for m=1 we build a graph G1
(t)
as follows
(Figure 5.5):
(1) Start with G1
(0)
, corresponding to an empty graph with no
nodes.
(2) Given G1
(t-1)
generate G1
(t)
by adding the node vt
and a single
link between vt
and vi
, where vi
is chosen with probability
That is, we place a link from the new node vt
to node vi
with prob-
ability ki
/(2t-1), where the new link already contributes to the de-
gree of vt
. Consequently node vt
can also link to itself with prob-
ability 1/(2t - 1), the second term in (5.2). Note also that the model
permits self-loops and multi-links. Yet, their number becomes
negligible in the t→∞ limit.
For m > 1 we build Gm
(t)
by adding m links from the new node vt
one by one, in each step allowing the outward half of the newly
added link to contribute to the degrees.
(5.2)
The Linearized Chord Diagram (LCD)
The construction of the LCD, the version
of the Barabási-Albert model amenable to
exact mathematical calculations [10]. The
figure shows the first four steps of the net-
work's evolution for m=1:
G1
(0)
: We start with an empty network.
G1
(1)
: The first node can only link to itself,
forming a self-loop. Self-loops are allowed,
and so are multi-links for m>1.
G1
(2)
: Node 2 can either connect to node 1
with probability 2/3, or to itself with prob-
ability 1/3. According to (5.2), half of the
links that the new node 2 brings along is
already counted as present. Consequently
node 1 has degree k1
=2 at node 2 has degree
k2
=1, the normalization constant being 3.
G1
(3)
: Let us assume that the first of the two
G1
(t)
network possibilities have material-
ized. When node 3 comes along, it again
has three choices: It can connect to node 2
with probability 1/5, to node 1 with proba-
bility 3/5 and to itself with probability 1/5.
Figure 5.5
p =
ki
2t 1
if 1 i t 1
1
2t 1
, if i = t
1
1 2 or 1 2
1 2
3
or
1
2
3
or
1 2
3
p=
2
3
p=
1
3
p=
1
5
p=
3
5
p=
1
5
G1
(3)
G1
(0)
G1
(1)
G1
(2)

DEGREE DYNAMICS
SECTION 5.3
To understand the emergence of the scale-free property, we need to fo-
cus on the time evolution of the Barabási-Albert model. We begin by ex-
ploring the time-dependent degree of a single node [11].
In the model an existing node can increase its degree each time a new
node enters the network. This new node will link to m of the N(t) nodes
already present in the system. The probability that one of these links con-
nects to node i is given by (5.1).
Let us approximate the degree ki
with a continuous real variable, repre-
senting its expectation value over many realizations of the growth process.
The rate at which an existing node i acquires links as a result of new nodes
connecting to it is
The coefficient m describes that each new node arrives with m links.
Hence, node i has m chances to be chosen. The sum in the denominator of
(5.3) goes over all nodes in the network except the newly added node, thus
Therefore (5.4) becomes
For large t the (-1) term can be neglected in the denominator, obtaining
By integrating (5.6) and using the fact that ki
(ti
)=m, meaning that node i
joins the network at time ti
with m links, we obtain
∑
= Π =
=
−
dk
dt
m k m
k
k
( ) .
i
i
i
j
N
j
1
1
(5.3)
(5.4)
(5.5)
(5.6)
k mt m
2 .
j
N
j
1
1
∑ = −
=
−
=
−
dk
dt
k
t
2 1
i i
dk
k
dt
t
1
2
i
i
=
.
.

12
We call β the dynamical exponent and has the value
Equation (5.7) offers a number of predictions:
• The degree of each node increases following a power-law with the
same dynamical exponent β =1/2 (Figure 5.6a). Hence all nodes follow
the same dynamical law.
• The growth in the degrees is sublinear (i.e. β < 1). This is a consequence
of the growing nature of the Barabási-Albert model: Each new node has
more nodes to link to than the previous node. Hence, with time the ex-
isting nodes compete for links with an increasing pool of other nodes.
• The earlier node i was added, the higher is its degree ki
(t). Hence, hubs
are large because they arrived earlier, a phenomenon called first-mov-
er advantage in marketing and business.
• The rate at which the node i acquires new links is given by the deriva-
tive of (5.7)
indicating that in each time step older nodes acquire more links (as
they have smaller ti
). Furthermore the rate at which a node acquires
links decreases with time as t−1/2
. Hence, fewer and fewer links go to a
node.
In summary, the Barabási-Albert model captures the fact that in real
networks nodes arrive one after the other, offering a dynamical descrip-
tion of a network’s evolution. This generates a competition for links during
which the older nodes have an advantage over the younger ones, eventual-
ly turning into hubs.
1
2
β =
dk t
dt
m
t t
( )
2
1
.
i
i
= (5.8)
DEGREE DYNAMICS
,
BOX 5.2
TIME IN NETWORKS
As we compare the predictions of the
network models with real data, we
have to decide how to measure time
in networks. Real networks evolve
over rather different time scales:
World Wide Web
The first webpage was created in
1991. Given its trillion documents,
the WWW added a node each milli-
second (103
sec).
Cell
The cell is the result of 4 billion years
of evolution. With roughly 20,000
genes in a human cell, on average
the cellular network added a node
every 200,000 years (~1013
sec).
Given these enormous time-scale
differences it is impossible to use
real time to compare the dynamics
of different networks. Therefore, in
network theory we use event time,
advancing our time-step by one each
time when there is a change in the
network topology.
For example, in the Barabási-Albert
model the addition of each new node
corresponds to a new time step,
hence t=N. In other models time
is also advanced by the arrival of a
new link or the deletion of a node.
If needed, we can establish a direct
mapping between event time and
the physical time.
t m
2 .
ki (t) = m
t
ti






β
t m
2 . (5.7)

13
(a) The growth of the degrees of nodes added
at time t =1, 10, 102
, 103
, 104
, 105
(continuous
lines from left to right) in the Barabási-Albert
model. Each node increases its degree follow-
ing (5.7). Consequently at any moment the old-
er nodes have higher degrees. The dotted line
corresponds to the analytical prediction (5.7)
with β = 1/2.
(b) Degree distribution of the network after
adding N = 102
, 104
, and 106
nodes, i.e. at time
t = 102
, 104
, and 106
(illustrated by arrows in
(a)). The larger the network, the more obvious
is the power-law nature of the degree distri-
bution. Note that we used linear binning for
pk
to better observe the gradual emergence of
the scale-free state.
Figure 5.6
Degree Dynamics
(a)
(b)
DEGREE DYNAMICS
t
k
N=106
tβ
k
SINGLE NETWORK
101
100
100
100
101
102
103
104
105
106
101
102
103
104
k
N=104
100
100
101
102
103
104
105
106
101
102
103
104
k
pk
pk
pk
N=102
100
100
101
102
103
104
105
106
101
102
103
104
100
101
102
103
104
105
103
102
104
106
105
< < <

DEGREE DISTRIBUTION
SECTION 5.4
The distinguishing feature of the networks generated by the Barabá-
si-Albert model is their power-law degree distribution (Figure 5.4). In this
section we calculate the functional form of pk
, helping us understand its
origin.
A number of analytical tools are available to calculate the degree distri-
bution of the Barabási-Albert network. The simplest is the continuum theo-
ry that we started developing in the previous section [1, 11]. It predicts the
degree distribution (BOX 5.3),
with
Therefore the degree distribution follows a power law with degree ex-
ponent γ=3, in agreement with the numerical results (Figures 5.4 and 5.7).
Moreover (5.10) links the degree exponent, γ, a quantity characterizing the
network topology, to the dynamical exponent β that characterizes a node’s
temporal evolution, revealing a deep relationship between the network's
topology and dynamics.
While the continuum theory predicts the correct degree exponent, it
fails to accurately predict the pre-factors of (5.9). The correct pre-factors
can be obtained using a master [12] or rate equation [13] approach or cal-
culated exactly using the LCD model [10] (BOX 5.2). Consequently the exact
degree distribution of the Barabási-Albert model is (ADVANCED TOPICS 5.A)
Equation (5.11) has several implications:
• For large k (5.11) reduces to pk
~ k-3
, or γ = 3, in line with (5.9) and (5.10).
• The degree exponent γ is independent of m, a prediction that agrees
(5.9)
(5.10)
(5.11)
p k m k
( ) 2 1/
≈ β γ
−
1
1 3.
γ
β
= + =
p
m m
k k k
2 ( 1)
( 1)( 2)
k =
+
+ +
(a) We generated networks with N=100,000
and m0
=m=1 (blue), 3 (green), 5 (grey), and 7
(orange). The fact that the curves are parallel
to each other indicates that γ is independent
of m and m0
. The slope of the purple line is -3,
corresponding to the predicted degree expo-
nent γ=3. Inset: (5.11) predicts pk
~2m2
, hence
pk
/2m2
should be independent of m. Indeed,
by plotting pk
/2m2
vs. k, the data points shown
in the main plot collapse into a single curve.
(b) The Barabási-Albert model predicts that
pk
is independent of N. To test this we plot pk
for N = 50,000 (blue), 100,000 (green), and
200,000 (grey), with m0
=m=3. The obtained pk
are practically indistinguishable, indicating
that the degree distribution is stationary, i.e.
independent of time and system size.
Figure 5.7
Probing the Analytical Predictions
(a)
(b)
.
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
k
k
100
101
102
103
104
pk
pk
/2m2
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
k
100
101
102
103
104
pk

15
THE BARABÁSI-ALBERT MODEL DEGREE DISTRIBUTION
with the numerical results (Figure 5.7a).
• The power-law degree distribution observed in real networks de-
scribes systems of rather different age and size. Hence, an approriate
model should lead to a time-independent degree distribution. Indeed,
according to (5.11) the degree distribution of the Barabási-Albert mod-
el is independent of both t and N. Hence the model predicts the emer-
gence of a stationary scale-free state. Numerical simulations support
this prediction, indicating that pk
observed for different t (or N) fully
overlap (Figure 5.7b).
• Equation (5.11) predicts that the coefficient of the power-law distribu-
tion is proportional to m(m + 1) (or m2
for large m), again confirmed by
numerical simulations (Figure 5.7a, inset).
In summary, the analytical calculations predict that the Barabási-Al-
bert model generates a scale-free network with degree exponent γ=3. The
degree exponent is independent of the m and m0
parameters. Further-
more, the degree distribution is stationary (i.e. time invariant), explaining
why networks with different history, size and age develop a similar degree
distribution.
BOX 5.3
CONTINUUM THEORY
To calculate the degree distribu-
tion of the Barabási-Albert mod-
el in the continuum approxi-
mation we first calculate the
number of nodes with degree
smaller than k, i.e. ki
(t) < k. Using
(5.7), we write
In the model we add a node at
equal time step (BOX 5.2). There-
fore the number of nodes with
degree smaller than k is
Altogether there are N=m0
+t
nodes, which becomes N≈t in the
large t limit. Therefore the prob-
ability that a randomly chosen
node has degree k or smaller,
which is the cumulative degree
distribution, follows
By taking the derivative of (5.14)
we obtain the degree distribu-
tion
which is (5.9).
ti < t
m
k
⎛
⎝
⎜
⎞
⎠
⎟
1/β
t
m
k
.
1/






β
P(k) = 1−
m
k
⎛
⎝
⎜
⎞
⎠
⎟
1/β
.
(5.12)
(5.13)
(5.14)
(5.15)
p
P k
k
m
k
m k
( ) 1
2 ,
k
1/
1/ 1
2 3
β
=
∂
∂
= =
β
β+
−
.
,

THE ABSENCE OF GROWTH
OR PREFERENTIAL ATTACHMENT
SECTION 5.5
The coexistence of growth and preferential attachment in the Barabá-
si-Albert model raises an important question: Are they both necessary for
the emergence of the scale-free property? In other words, could we gener-
ate a scale-free network with only one of the two ingredients? To address
these questions, next we discuss two limiting cases of the model, each con-
taining only one of the two ingredients [1, 11].
MODEL A
To test the role of preferential attachment we keep the growing charac-
ter of the network (ingredient A) and eliminate preferential attachment
(ingredient B). Hence, Model A starts with m0
nodes and evolves follow-
ing these steps:
(A) Growth
At each time step we add a new node with m(≤m0
) links that connect
to m nodes added earlier.
The probability that a new node links to a node with degree ki
is
That is, Π(ki
) is independent of ki
, indicating that new nodes choose ran-
domly the nodes they link to.
The continuum theory predicts that for Model A ki
(t) increases logarith-
mically with time
a much slower growth than the power law increase (5.7). Consequently
the degree distribution follows an exponential (Figure 5.8a)
k
m t
( )
1
( 1)
.
i
0
Π =
+ − (5.16)
(5.17)
ki (t) = mln
m0 + t −1
m0 + ti −1
⎛
⎝
⎜
⎞
⎠
⎟ ,

17
An exponential function decays much faster than a power law, hence
it does not support hubs. Therefore the lack of preferential attachment
eliminates the network’s scale-free character and the hubs. Indeed, as
all nodes acquire links with equal probabilty, we lack a rich-get-richer
process and no clear winner can emerge.
MODEL B
To test the role of growth next we keep preferential attachment (ingre-
dient B) and eliminate growth (ingredient A). Hence, Model B starts with
N nodes and evolves following this step:
At each time step a node is selected randomly and connected to node i
with degree ki
already present in the network, where i is chosen with
probability Π(k). As Π(0)=0 nodes with k=0 are assumed to have k=1, oth-
erwise they can not acquire links.
In Model B the number of nodes remains constant during the net-
work’s evolution, while the number of links increases linearly with
time. As a result for large t the degree of each node also increases
linearly with time (Figure 5.7b, inset)
Indeed, in each time step we add a new link, without changing the
number of nodes.
At early times, when there are only a few links in the network (i.e. L
≪ N), each new link connects previously unconnected nodes. In this
stage the model’s evolution is indistinguishable from the Barabá-
si-Albert model with m=1. Numerical simulations show that in this
regime the model develops a degree distribution with a power-law
tail (Figure 5.8b).
Yet, pk
is not stationary. Indeed, after a transient period the node de-
grees converge to the average degree (5.19) and the degree develops a
peak (Figure 5.8b). For t → N(N-1)/2 the network becomes a complete
graph in which all nodes have degree kmax
=N-1, hence pk
= δ(N-1).
In summary, the absence of preferential attachment leads to a growing
network with a stationary but exponential degree distribution. In contrast
the absence of growth leads to the loss of stationarity, forcing the network
to converge to a complete graph. This failure of Models A and B to reproduce
the empirically observed scale-free distribution indicates that growth and
preferential attachment are simultaneously needed for the emergence of the
scale-free property.
(5.19)
Numerical simulations probing the role of
growth and preferential attachment.
(a) Model A
Degree distribution for Model A, that in-
corporates growth but lacks preferential
attachment. The symbols correspond to
m0
=m=1 (circles), 3 (squares), 5 diamonds),
7 (triangles) and N=800,000. The linear-log
plot indicates that the resulting network
has an exponential pk
, as predicted by (5.18).
Inset: Time evolution of the degree of two
nodes added at t1
=7 and t2
=97 for m0
=m=3.
The dashed line follows (5.17).
(b) Model B
Degree distribution for Model B, that
lacks growth but incorporates preferen-
tial attachment, shown for N=10,000 and
t=N (circles), t=5N (squares), and t=40N
(diamonds). The changing shape of pk
in-
dicates that the degree distribution is not
stationary.
Inset: Time dependent degrees of two
nodes (N=10,000), indicating that ki
(t)
grows linearly, as predicted by (5.19). Af-
ter [11].
Figure 5.8
Model A and Model B
(a)
(b)
THE ABSENCE OF GROWTH
OR PREFERENTIAL ATTACHMENT
ki (t)
2
N
t .
p(k)=
e
m
exp
k
m
(5.18)
. 10-1
10-3
10-5
101
103
105
10-7
0 50 100
100
100
101
102
103
50000
0
10-1
10-2
10-3
10-4
10-5
pk
k
pk
k
MODEL A
MODEL B
t
t
0
10
20
10
0
20
30
40
ki
(t)
ki
(t)

MEASURING PREFERENTIAL
ATTACHMENT
SECTION 5.6
In the previous section we showed that growth and preferential attach-
ment are jointly responsible for the scale-free property. The presence of
growth in real systems is obvious: All large networks have reached their
current size by adding new nodes. But to convince ourselves that preferen-
tial attachment is also present in real networks, we need to detect it experi-
mentally. In this section we show how to detect preferential attachment by
measuring the Π(k) function in real networks.
Preferential attachment relies on two distinct hypotheses:
Hypothesis 1
The likelihood to connect to a node depends on that node’s degree k.
This is in contrast with the random network model, for which Π(k) is
independent of k.
Hypothesis 2
The functional form of Π(k) is linear in k.
Both hypotheses can be tested by measuring Π(k). We can determine Π(k)
for systems for which we know the time at which each node joined the net-
work, or we have at least two network maps collected at not too distant
moments in time [14, 15].
Consider a network for which we have two different maps, the first taken
at time t and the second at time t + Δt (Figure 5.9a). For nodes that changed
their degree during the Δt time frame we measure Δki
= ki
(t+Δt )−ki
(t). Ac-
cording to (5.1), the relative change Δki
/Δt should follow
providing the functional form of preferential attachment. For (5.20) to be
valid we must keep Δt small, so that the changes in Δk are modest. But Δt
must not be too small so that there are still detectable differences between
the two networks.
Figure 5.9
Detecting Preferential Attachment
(a) If we have access to two maps of the same
network taken at time t and t+Δt, compar-
ing them allows us to measure the Π(k)
function. Specifically, we look at nodes
that have gained new links thanks to the
arrival of the two new green nodes at t+Δt.
The orange lines correspond to links that
connect previously disconnected nodes,
called internal links. Their role is discussed
in CHAPTER 6.
(b) In the presence of preferential attachment
Δk/Δt will depend linearly on a node’s de-
gree at time t.
(c) The scaling of the cumulative preferential
attachment function π(k) helps us detect
the presence or absence of preferential at-
tachment (Figure 5.10).
(5.20)
Δk
Δt
k
( )
i
i
~ Π
(
k)
t + ∆ t
t
No PA
N
o
P
A
~
k
P
A
~
k
2
k k
P
A
~
Π
π
(
k)
∆
k
∆
t
(a)
(b) (c)
,

In practice the obtained Δki
/Δt curve can be noisy. To reduce this noise
we measure the cumulative preferential attachment function
In the absence of preferential attachment we have Π(ki
)=constant, hence,
π(k) ∼ k according to (5.21). If linear preferential attachment is present, i.e.
if Π(ki
)=ki
, we expect π(k) ∼ k2
.
Figure 5.10 shows the measured π(k) for four real networks. For each sys-
tem we observe a faster than linear increase in π(k), indicating the pres-
ence of preferential attachment. Figure 5.10 also suggests that Π(k) can be
approximated with
For the Internet and citation networks we have α ≈ 1, indicating that
Π(k) depends linearly on k, following (5.1). This is in line with Hypotheses
1 and 2. For the co-authorship and the actor network the best fit provides
α=0.9±0.1 indicating the presence of a sublinear preferential attachment.
In summary, (5.20) allows us to detect the presence (or absence) of pref-
erential attachment in real networks. The measurements show that the at-
tachment probability depends on the node degree. We also find that while
in some systems preferential attachment is linear, in others it can be sub-
linear. The implications of this non-linearity are discussed in the next sec-
tion.
(5.21)
(5.22)
The figure shows the cumulative preferential
attachment function π(k), defined in (5.21), for
several real systems:
(a) Citation network.
(b) Internet.
(c) Scientific collaboration network (neurosci-
ence).
(d) Actor network.
In each panel we have two lines to guide the
eye: The dashed line corresponds to linear
preferential attachment (π(k)∼k2
) and the
continuous line indicates the absence of pref-
erential attachment (π(k)∼k). In line with Hy-
pothesis 1 we detect a k-dependence in each
dataset. Yet, in (c) and (d) π(k) grows slower
than k2
, indicating that for these systems pref-
erential attachment is sublinear, violating
Hypothesis 2. Note that these measurements
only consider links added through the arrival
of new nodes, ignoring the addition of inter-
nal links. After [14].
Figure 5.10
Evidence of Preferential Attachment
CITATION
NETWORK
INTERNET
COLLABORATION
NETWORK
ACTOR
NETWORK
(a)
(c)
(b)
(d)
MEASURING PREFERENTIAL
ATTACHMENT IN REAL NETWORKS
k k
( )
Π α
∼
.
.
101
102
101
k k
k k
π(k)
π(k) π(k)
π(k)
100
102
103
104
100
101
102
103
100
101
102
103
100
102
10-2
10-4
100
102
10-2
10-4
10-6
100
102
10-2
10-4
10-6
100
102
10-2
10-4
10-6
k k
( ) ( )
k
k
i
0
i
∑
π = Π
=

NON-LINEAR PREFERENTIAL
ATTACHMENT
SECTION 5.7
(5.23)
The observation of sublinear preferential attachment in Figure 5.10 raises
an important question: What is the impact of this nonlinearity on the net-
work topology? To answer this we replace the linear preferential attach-
ment (5.1) with (5.22) and calculate the degree distribution of the obtained
nonlinear Barabási-Albert model.
The behavior for α=0 is clear: In the absence of preferential attachment
we are back to Model A discussed in SECTION 5.4. Consequently the degree
distribution follows the exponential (5.17).
For α = 1 we recover the Barabási-Albert model, obtaining a scale-free
network with degree distribution (5.14).
Next we focus on the case α ≠ 0 and α ≠ 1. The calculation of pk
for an ar-
bitrary α predicts several scaling regimes [13] (ADVANCED TOPICS 5.B):
Sublinear Preferential Attachment (0 < α < 1)
For any α > 0 new nodes favor the more connected nodes over the less
connected nodes. Yet, for α < 1 the bias is weak, not sufficient to generate
a scale-free degree distribution. Instead, in this regime the degrees
follow the stretched exponential distribution (SECTION 4.10)
,
where μ(α) depends only weakly on α. The exponential cutoff in (5.23)
implies that sublinear preferential attachment limits the size and the
number of the hubs.
Sublinear preferential attachment also alters the size of the largest de-
gree, kmax
. For a scale-free network kmax
scales polynomially with time,
following (4.18). For sublinear preferential attachment we have
, (5.24)
105
104
103
102
101
100
k
~(lnt)2
~t1/2
~t
α=0.5
α=1
α=2.5
100
101
102
103
104
105
kmax
Figure 5.11
The Growth of the Hubs
The nature of preferential attachment affects
the degree of the largest node. While in a scale-
free network (α=1) the biggest hub grows as t1/2
(green curve, (4.18)), for sublinear preferential
attachment (α<1) this dependence becomes
logarithmic, following (5.24). For superlinear
preferential attachment (α>1) the biggest hub
grows linearly with time, always grabbing a
finite fraction of all links, following (5.25). The
symbols are provided by numerical simula-
tions; the dotted lines represent the analytical
predictions.
kmax ∼ (lnt)1/(1−α)
pk k exp
-2µ( )
k (1 )
k1

a logarithmic dependence that predicts a much slower growth of the
maximum degee than the polynomial. This slower growth is the reason
why the hubs are smaller for α < 1 (Figure 5.11).
Superlinear Preferential Attachment (α > 1)
For α > 1 the tendency to link to highly connected nodes is enhanced,
accelerating the rich-gets-richer process. The consequence of this is most
obvious for α > 2, when the model predicts a winner-takes-all phenom-
enon: almost all nodes connect to a few super-hubs. Hence we observe
the emergence of a hub-and-spoke network, in which most nodes link
directly to a few central nodes. The situation for 1 < α < 2 is less extreme,
but similar.
This winner-takes-all process alters the size of the largest hub as well,
finding that (Figure 5.11).
Hence for α > 1 the largest hub links to a finite fraction of nodes in the
system.
In summary, nonlinear preferential attachment changes the degree
distribution, either limiting the size of the hubs (α < 1), or leading to su-
per-hubs (α > 1, Figure 5.12). Consequently, Π(k) needs to depend strictly lin-
early on the degrees for the resulting network to have a pure power law pk
.
While in many systems we do observe such a linear dependence, in others,
like the scientific collaboration network and the actor network, preferen-
tial attachment is sublinear. This nonlinear Π(k) is one reason the degree
distribution of real networks deviates from a pure power-law. Hence for
systems with sublinear Π(k) the stretched exponential (5.23) should offer a
better fit to the degree distribution.
(5.25)
21
THE BARABÁSI-ALBERT MODEL NON-LINEAR PREFERENTIAL ATTACHMENT
k t
max  .

22
The scaling regimes characterizing the non-
linear Barabási-Albert model. The three top
panels show pk
for different α (N=104
). The
network maps show the corresponding topol-
ogies (N=100). The theoretical results predict
the existence of four scaling regimes:
No Preferential Attachment (α=0)
The network has a simple exponential degree
distribution, following (5.18). Hubs are absent
and the resulting network is similar to a ran-
dom network.
Sublinear Regime (0<α<1)
The degree distribution follows the stretched
exponential (5.23), resulting in fewer and
smaller hubs than in a scale-free network. As
α → 1 the cutoff length increases and pk
fol-
lows a power law over an increasing range of
degrees.
Linear Regime (α=1)
This corresponds to the Barabási-Albert mod-
el, hence the degree distribution follows a
power law.
Superlinear Regime (α>1)
The high-degree nodes are disproportionately
attractive. A winner-takes-all dynamics leads
to a hub-and-spoke topology. In this configu-
ration the earliest nodes become super hubs
and all subsequent nodes link to them. The
degree distribution, shown for α=1.5 indicates
the coexistence of many small nodes with a
few super hubs in the vicinity of k=104
.
Figure 5.12
Nonlinear Preferential Attachment
NON-LINEAR PREFERENTIAL ATTACHMENT
SUBLINEAR
EXPONENTIAL
RANDOM
NO PREFERENTIAL
ATTACHMENT
NETWORK
RANDOM
SUBLINEAR SUPERLINEAR
LINEAR
SCALE-FREE
m WINNERS
TAKE ALL
�(k) ~ kα
~ lnt ~ (lnt) ~ t
(α=1) (α>1)
(0<α<1)
kmax
pk
POWER LAW
HUB-AND-SPOKE
TOPOLOGY
STRETCHED
EXPONENTIAL
SUPERLINEAR
LINEAR
0.5
10-6
10-5
10-4
10-3
10-1
100
100
102
103
104
101
k
1.5 α
=
1
2
α =
3
2
α
=1
α
pk
10-6
10-5
10-4
10-3
10-1
100
100
102
103 104
101
k
pk
10-6
10-5
10-4
10-3
10-1
100
100
102
103
104
101
k
pk
1
0
1
1
t
1
1

THE ORIGINS OF PREFERENTIAL
ATTACHMENT
SECTION 5.8
Given the key role preferential attachment plays in the evolution of
real networks, we must ask, where does it come from? The question can be
broken to two narrower issues:
Why does Π(k) depend on k?
Why is the dependence of Π(k) linear in k?
In the past decade we witnessed the emergence of two philosophically
different answers to these questions. The first views preferential attach-
ment as the interplay between random events and some structural prop-
erty of a network. These mechanisms do not require global knowledge of
the network but rely on random events, hence we will call them local or
random mechanisms. The second assumes that each new node or link bal-
ances conflicting needs, hence they are preceeded by a cost-benefit anal-
ysis. These models assume familiarity with the whole network and rely
on optimization principles, prompting us to call them global or optimized
mechanisms. In this section we discuss both approaches.
LOCAL MECHANISMS
The Barabási-Albert model postulates the presence of preferential at-
tachment. Yet, as we show below, we can build models that generate scale-
free networks apparently without preferential attachment. They work by
generating preferential attachment. Next we discuss two such models and
derive Π(k) for them, allowing us to understand the origins of preferential
attachment.
Link Selection Model
The link selection model offers perhaps the simplest example of a local
mechanism that generates a scale-free network without preferential
attachment [16]. It is defined as follows (Figure 5.13):
• Growth: At each time step we add a new node to the network.
• Link Selection: We select a link at random and connect the new node to
Figure 5.13
Link Selection Model
(a)
(b)
(a) The network grows by adding a new node,
that selects randomly a link from the network
(shown in purple).
(b) The new node connects with equal prob-
ability to one of the two nodes at the ends of
the selected link. In this case the new node
connected to the node at the right end of the
selected link.
NEW NODE

24
one of the two nodes at the two ends of the selected link.
The model requires no knowledge about the overall network topology,
hence it is inherently local and random. Unlike the Barabási-Albert
model, it lacks a built-in Π(k) function. Yet, next we show that it gener-
ates preferential attachment.
We start by writing the probability qk
that the node at the end of a ran-
domly chosen link has degree k as
Equation (5.26) captures two effects:
• The higher is the degree of a node, the higher is the chance that it is
located at the end of the chosen link.
• The more degree-k nodes are in the network (i.e., the higher is pk
),
the more likely that a degree k node is at the end of the link.
In (5.26) C can be calculated using the normalization condition Σqk
= 1,
obtaining C=1/⟨k⟩. Hence the probability to find a degree-k node at the end
of a randomly chosen link is
Equation (5.27) is the probability that a new node connects to a node with
degree k. The fact that the bias in (5.27) is linear in k indicates that the link
selection model builds a scale-free network by generating linear preferen-
tial attachment.
Copying Model
While the link selection model offers the simplest mechanism for prefer-
ential attachment, it is neither the first nor the most popular in the class of
models that rely on local mechanisms. That distinction goes to the copying
model (Figure 5.14). The model mimics a simple phenomena: The authors of
a new webpage tend to borrow links from other webpages on related topics
[17, 18]. It is defined as follows:
In each time step a new node is added to the network. To decide where it
connects we randomly select a node u, corresponding for example to a web
document whose content is related to the content of the new node. Then we
follow a two-step procedure (Figure 5.14):
(i) Random Connection: With probability p the new node links to u,
which means that we link to the randomly selected web document.
(ii) Copying: With probability 1-p we randomly choose an outgoing
link of node u and link the new node to the link’s target. In other
words, the new webpage copies a link of node u and connects to its
target, rather than connecting to node u directly.
k =
kpk
〈k〉
The main steps of the copying model. A new
node connects with probability p to a randomly
chosen target node u, or with probability 1-p to
one of the nodes the target u points to. In other
words, with probabilty 1-p the new node copies a
link of its target u.
Figure 5.14
Copying Model
THE ORIGINS OF PREFERENTIAL ATTACHMENT
(5.27)
.
q Ckp
k k
= . (5.26)
NEW NODE
EXISTING
NETWORK
CHOOSE TARGET
TARGET
p 1-p
CHOOSE ONE OF THE
OUTGOING LINKS OF TARGET
u u u

25
The probability of selecting a particular node in step (i) is 1/N. Step (ii) is
equivalent with selecting a node linked to a randomly selected link. The
probability of selecting a degree-k node through this copying step (ii) is
k/2L for undirected networks. Combining (i) and (ii), the likelihood that a
new node connects to a degree-k node follows
which, being linear in k, predicts a linear preferential attachment.
The popularity of the copying model lies in its relevance to real systems:
• Social Networks: The more acquaintances an individual has, the
higher is the chance that she will be introduced to new individuals
by her existing acquaintances. In other words, we "copy" the friends
of our friends. Consequently without friends, it is difficult to make
new friends.
• Citation Networks: No scientist can be familiar with all papers pub-
lished on a certain topic. Authors decide what to read and cite by
"copying" references from the papers they have read. Consequently
papers with more citations are more likely to be studied and cited
again.
• Protein Interactions: Gene duplication, responsible for the emer-
gence of new genes in a cell, can be mapped into the copying model,
explaining the scale-free nature of protein interaction networks [19,
20].
Taken together, we find that both the link selection model and the copy-
ing model generate a linear preferential attachment through random
linking.
OPTIMIZATION
A longstanding assumption of economics is that humans make rational
decisions, balancing cost against benefits. In other words, each individ-
ual aims to maximize its personal advantage. This is the starting point
of rational choice theory in economics [21] and it is a hypothesis central
to modern political science, sociology, and philosophy. As we show be-
low, such rational decisions can lead to preferential attachment [22, 23,
24].
Consider the Internet, whose nodes are routers connected via cables.
Establishing a new Internet connection between two routers requires
us to lay down a new cable between them. As this is costly, each new link
is preceded by a careful cost-benefit analysis. Each new router (node)
will choose its link to balance access to good network performance (i.e.
proper bandwith) with the cost of laying down a new cable (i.e. physical
distance). This can be a conflicting desire, as the closest node may not
offer the best network performance.
(a) A small network, where the hj
term in the
cost function (5.28) is shown for each node.
Here hj
represents the network-based distance
of node j from node i=0, designated as the
"center" of the network, offering the best net-
work performance. Hence h0
=0 and h3
=2.
(b) A new node (green) will choose the node j
to which it connects by minimizing Cj
of (5.28).
(c)-(e) If δ is small the new node will connect
to the central node with hj
=0. As we increase
δ, the balance in (5.28) shifts, forcing the new
node to connect to more distant nodes. The
panels (c)-(e) show the choice of the new green
node makes for different values of δ.
(f) The basin of attraction for each node for
δ=10. A new node arriving inside a basin will
always link to the node at the center of the
basin. The size of each basin depends on the
degree of the node at its center. Indeed, the
smaller is hj
, the larger can be the distance to
the new node while still minimizing (5.28). Yet,
the higher is the degree of node j, the smaller
is its expected distance to the central node hj
.
Figure 5.15
Optimization Model
(a)
(c)
(e)
(b)
(d)
(f)
BASIN OF ATTRACTION
δ=10
t+1
t
δ=1000
δ=10
min{δdj5
+h5
}
d0
=0.49 d1
=0.57
d3
=0.31
d2
=0.76
h1
=1
h2
=1
h0
=0
h3
=2
h4
=1
j
δ=0.1
d4
=0.4
(k) =
p
N
+
1 p
2L
k ,

For simplicity let us assume that all nodes are located on a continent
with the shape of a unit square. At each time step we add a new node
and randomly choose a point within the square as its physical location.
When deciding where to connect the new node i, we calculate the cost
function [22]
which compares the cost of connecting to each node j already in the net-
work. Here dij
is the Euclidean distance between the new node i and the
potential target j, and hj
is the network-based distance of node j to the
first node of the network, which we designate as the desireable “center
“ of the network (Figure 5.15), offering the best network performance.
Hence hj
captures the “resources” offered by node j, measured by its
distance to the network’s center.
The calculations indicate the emergence of three distinct network to-
pologies, depending on the value of the parameter δ in (5.28) and N (Fig-
ure 5.15):
Star Network δ < (1/2)1/2
For δ = 0 the Euclidean distances are irrelevant, hence each node links
to the central node, turning the network into a star. We have a star con-
figuration each time when the hj
term dominates over δdij
in (5.28).
Random Network δ ≥ N1/2
For very large δ the contribution provided by the distance term δdij
overwhelms hj
in (5.28). In this case each new node connects to the node
closest to it. The resulting network will have a bounded degree distribu-
tion, like a random network (Figure 5.16b).
Scale-free Network 4 ≤ δ ≤ N1/2
Numerical simulations and analytical calculations indicate that for
intermediate δ values the network develops a scale-free topology [22].
The origin of the power law distribution in this regime is rooted in two
competing mechanisms:
(i) Optimization: Each node has a basin of attraction, so that nodes
landing in this basin will always link to it. The size of each basin cor-
relates with hj
of node j at its center, which in turn correlates with
the node’s degree kj
(Figure 5.15f).
(ii) Randomness: We choose randomly the location of the new node,
ending in one of the N basins of attraction. The node with the larg-
est degree has largest basin of attraction, hence gains the most new
nodes and links. This leads to preferential attachment, as document-
ed in Figure 5.16d.
In summary, we can build models that do not have an explicit Π(k) func-
INTRODUCTION
26
C d h
min [ ]
i j ij j
δ
= + (5.28)

INTRODUCTION
27
tion built into their definition, yet they generate a scale-free network. As
we showed in this section, these work by inducing preferential attachment.
The mechanism responsible for preferential attachment can have two fun-
damentally different origins (Figure 5.17): it can be rooted in random pro-
cesses, like link selection or copying, or in optimization, when new nodes
balance conflicting criteria as they decide where to connect. Note that each
of the mechanisms discussed above lead to linear preferential attachment,
as assumed in the Barabási-Albert model. We are not aware of mechanisms
capable of generating nonlinear preferential attachment, like those dis-
cussed in SECTION 5.7.
The diversity of the mechanisms discussed in this section suggest that
linear preferential attachment is present in so many and so different sys-
tems precisely because it can come from both rational choice and random
actions [25]. Most complex systems are driven by processes that have a bit
of both. Hence luck or reason, preferential attachment wins either way.

28
(a) The three network classes generated by the
optimization model: star, scale-free, and ex-
ponential networks. The topology of the net-
work in the unmarked area is unknown.
The vertical boundary of the star configura-
tion is at δ=(1/2)1/2
. This is the inverse of the
maximum distance between two nodes on a
square lattice with unit length, over which
the model is defined. Therefore if δ < (1/2)1/2
,
for any new node δdij
<1 and the cost (5.28) of
connecting to the central node is Ci
= δdij
+0,
always lower than connecting to any other
node at the cost of f(i,j) = δdij
+1. Therefore for δ
< (1/2)1/2
all nodes connect to node 0, resulting
in a network dominated by a single hub (star-
and-spoke network (c)).
The oblique boundary of the scale-free regime
is δ = N1/2
. Indeed, if nodes are placed random-
ly on the unit square, then the typical distance
between neighbors decreases as N−1/2
. Hence,
if dij
~N−1/2
then δdij
≥hij
for most node pairs.
Typically the path length to the central node hj
grows slower than N (in small-world networks
hj
~log N, in scale-free networks hj
~lnlnN).
Therefore Ci
is dominated by the δdij
term and
the smallest Ci
is achieved by minimizing the
distance-dependent term. Note that strictly
speaking the transition only occurs in the N →
∞ limit. In the white regime we lack an analyt-
ical form for the degree distribution.
(b) Degree distribution of networks generated
in the three phases marked in (a) for N=104
.
(c) Typical topologies generated by the optimi-
zation model for selected δ values. Node size is
proportional to its degree.
(d) We used the method described in SEC-
TION 5.6 to measure the preferential attach-
ment function. Starting from a network with
N=10,000 nodes we added a new node and
measured the degree of the node that it con-
nected to. We repeated this procedure 10,000
times, obtaining Π(k). The plots document the
presence of linear preferential attachment in
the scale-free phase, but its absence in the star
and the exponential phases.
Figure 5.16
Scaling in the Optimization Model
(a)
(b)
(c)
(d)
N
101
STAR
NETWORK
EXPONENTIAL
NETWORK
SCALE-FREE EXPONENTIAL
STAR
pk
k k k
k k
100
100
9000 0 100 200 300 0 2 4 6 8 10 12
400
0
0.2
0.4
0.8
1
10000 11000
104
100
101
102
k
101
102
103
103
100
101
102
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
100
pk
10-6
10-5
10-4
10-3
10-1
100
pk
10-5
10-4
10-3
10-1
100
105
103
102
δ
δ=0.1
Π(k)
0
0.01
0.02
0.03
Π(k)
0.0-100
5.0-10-5
1.0-10-4
1.5-10-4
2.0-10-4
Π(k)
δ=10 δ=1,000
SCALE-FREE
NETWORK

29
THE BARABÁSI-ALBERT MODEL THE ORIGINS OF PREFERENTIAL ATTACHMENT
The tension between randomness and opti-
mization, two apparently antagonistic ex-
planations for power laws, is by no means
new: In the 1960s Herbert Simon and Benoit
Mandelbrot have engaged in a fierce public
dispute over this very topic. Simon proposed
that preferential attachment is responsible
for the power-law nature of word frequencies.
Mandelbrot fiercely defended an optimiza-
tion-based framework. The debate spanned
seven papers and two years and is one of the
most vicious scientific disagreement on re-
cord.
In the context of networks today the argu-
ment titled in Simon’s favor: The power laws
observed in complex networks appear to be
driven by randomness and preferential at-
tachment. Yet, the optimization-based ideas
proposed by Mandelbrot play an important
role in explaining the origins of preferential
attachment. So at the end they were both
right.
FIG. 5.17
LUCK OR REASON: AN ANCIENT FIGHT
7
Mandelbrot publishes a
comment on Simon’s paper [27]
writing:
In a 19 page response entitled
Final Note, Mandelbrot
states [29]:
In the creatively titled Post
Scriptum to Final Note
Mandlebrot [31] writes
Simon’s final note ends but does
not resolve the debate [33]
5
Benoit
1953
1955
Benoit
Dr. Mandelbrot’s principal and mathemati-
cal objections to the model are shown to be
unfounded
The essence of Simon’s lengthy
reply a year later is well
summarized in its abstract [28].
Simon’s model is analytically circular...
1959
1961
1960
...Most of Simon’s (1960) reply was irrelevant.
My criticism has not changed since I first
had the privilege of commenting upon a
draft of Simon.
This present ‘Reply’ refutes the almost
entirely new arguments introduced by Dr.
Mandelbrot in his “Final Note”...
Simon’s subsequent Reply to
‘Final Note’ by Mandelbrot
does not concede [30]
1961
1961
Dr. Mandelbrot has proposed a new set of
objections to my 1955 models of Yule
distributions. Like earlier objections, these
are invalid. 1961
3
1
4
2
6
8
In On a Class of Skew Distribution Functions
Herbert Simon [6] proposes randomness
as the origin of power laws and dismisses
Mandelbrot’s claim that power law are
rooted in optimization.
In An Informal Theory of the Statistical
Structure of Languages [26] Benoit
Mandelbrot proposes optimization as the
origin of power laws.
Benoit
Simon’s final reply ends but
does not resolve the debate [32]

DIAMETER AND CLUSTERING
COEFFICIENT
SECTION 5.9
To complete the characterization of the Barabási-Albert model we dis-
cuss the behavior of the network diameter and the clustering coefficient.
Diameter
The network diameter, representing the maximum distance in the
Barabási-Albert network, follows for m > 1 and large N [33, 34]
Therefore the diameter grows slower than ln N, making the distances
in the Barabási-Albert model smaller than the distances observed in a
random graph of similar size. The difference is particularly relevant for
large N.
Note that while (5.29) is derived for the diameter, the average distance
〈d〉 scales in a similar fashion. Indeed, as we show in Figure 5.18, for small
N the ln N term captures the scaling of 〈d〉 with N, but for large N(≥104
)
the impact of the logarithmic correction ln ln N becomes noticeable.
Clustering coefficient
The clustering coefficient of the Barabási-Albert model follows (AD-
VANCED TOPICS 5.C) [35, 36]
The prediction (5.30) is quite different from the 1/N dependence ob-
tained for the random network model (Figure 5.19). The difference comes
in the (lnN)2
term, that increases the clustering coefficient for large N.
Consequently the Barabási-Albert network is locally more clustered
than a random network.
F
F
A
Clustering Coefficient
The dependence of the average distance on the
system size in the Barabási-Albert model. The
continuous line corresponds to the exact re-
sult (5.29), while the dotted line corresponds to
the prediction (3.19) for a random network. The
analytical predictions do not provide the exact
perfactors, hence the lines are not fits, but in-
dicate only the predicted N-dependent trends.
The results were averaged for ten independent
runs for m = 2.
The dependence of the average clustering co-
efficient on the system size N for the Barabá-
si-Albert model. The continuous line corre-
sponds to the analytical prediction (5.30), while
the dotted line corresponds to the prediction
for a random network, for which ⟨C⟩∼1/N. The
results are averaged for ten independent runs
for m = 2. The dashed and continuous curves
are not fits, but are drawn to indicate the pre-
dicted N dependent trends.
(5.29)
(5.30)
.
,
RANDOM
NETWORK
ln N
lnln N
N
~ln N
102
101 104
103 106
105
d
0
1
2
3
4
6
7
8
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
RANDOM
NETWORK
~1/ N
(lnN)2
N
N
C
d
N
N
ln
lnln
C
N
N
(ln )2
〈 〉 ∼

SUMMARY
SECTION 5.10
The most important message of the Barabási-Albert model is that net-
work structure and evolution are inseparable. Indeed, in the Erdős-Rényi,
Watts-Strogatz, the configuration and the hidden parameter models the
role of the modeler is to cleverly place the links between a fixed number of
nodes. Returning to our earlier analogy, the networks generated by these
models relate to real networks like a photo of a painting relates to the
painting itself: It may look like the real one, but the process of generating
a photo is drastically different from the process of painting the original
painting. The aim of the Barabási-Albert model is to capture the process-
es that assemble a network in the first place. Hence, it aims to paint the
painting again, coming as close as possible to the original brush strokes.
Consequently, the modeling philosophy behind the model is simple: to un-
derstand the topology of a complex system, we need to describe how it came
into being.
Random networks, the configuration and the hidden parameter models
will continue to play an important role as we explore how certain network
characteristics deviate from our expectations. Yet, if we want to explain
the origin of a particular network property, we will have to use models that
capture the system’s genesis.
The Barabási-Albert model raises a fundamental question: Is the com-
bination of growth and preferential attachment the real reason why net-
works are scale-free? We offered a necessary and sufficient argument to
address this question. First, we showed that growth and preferential at-
tachment are jointly needed to generate scale-free networks, hence if one
of them is absent, either the scale-free property or stationarity is lost. Sec-
ond, we showed that if they are both present, they do lead to scale-free net-
works. This argument leaves one possibility open, however: Do these two
mechanisms explain the scale-free nature of all networks? Could there
be some real networks that are scale-free thanks to some completely dif-
ferent mechanism? The answer is provided in SECTION 5.9, where we did
encountered the link selection, the copying and the optimization models
that do not have a preferential attachment function built into them, yet

32
THE BARABÁSI-ALBERT MODEL SUMMARY
they do lead to a scale-free network. We showed that they do so by generat-
ing a linear Π(k). This finding underscores a more general pattern: To date
all known models and real systems that are scale-free have been found to
have preferential attachment. Hence the basic mechanisms of the Barabá-
si-Albert model appear to capture the origin of their scale-free topology.
The Barabási-Albert model is unable to describe many characteristics
of real systems:
• The model predicts γ=3 while the degree exponent of real networks
varies between 2 and 5 (Table 4.2).
• Many networks, like the WWW or citation networks, are directed,
while the model generates undirected networks.
• Many processes observed in networks, from linking to already exist-
ing nodes to the disappearance of links and nodes, are absent from
the model.
• The model does not allow us to distinguish between nodes based on
some intrinsic characteristics, like the novelty of a research paper or
the utility of a webpage.
•While the Barabási-Albert model is occasionally used as a model of the
Internet or the cell, in reality it is not designed to capture the details of
any particular real network. It is a minimal, proof of principle model
whose main purpose is to capture the basic mechanisms responsible
for the emergence of the scale-free property. Therefore, if we want to
understand the evolution of systems like the Internet, the cell or the
WWW, we need to incorporate the important details that contribute
to the time evolution of these systems, like the directed nature of the
WWW, the possibility of internal links and node and link removal.
As we show in CHAPTER 6, these limitations can be systematically re-
solved.
BOX 5.5
AT A GLANCE:
BARABÁSI-ALBERT MODEL
Number of Nodes
N = t
Number of Links
N = mt
Average Degree
⟨k⟩ = 2m
Degree Dynamics
ki
(t) = m (t/ti
)β
Dynamical Exponent
β = 1/2
Degree Distribution
pk
∼ k-γ
Degree Exponent
γ = 3
Average Distance
Clustering Coefficient
⟨C⟩ ∼ (lnN)2
/N
lnN
ln lnN
d ~

SECTION 5.11
HOMEWORK
5.1. Generating Barabási-Albert Networks
With the help of a computer, generate a network with N = 104
nodes
using the Barabási-Albert model with m = 4. Use as initial condition a fully
connected network with m = 4 nodes.
(a) Measure the degree distribution at intermediate steps, namely
when the network has 102
, 103
and 104
nodes.
(b) Compare the distributions at these intermediate steps by plot-
ting them together and fitting each to a power-law with degree
exponent γ. Do the distributions converge?
(c) Plot together the cumulative degree distributions at intermedi-
ate steps.
(d) Measure the average clustering coefficient in function of N.
(e) Following Figure 5.6a, measure the degree dynamics of one of the
initial nodes and of the nodes added to the network at time t =
100, t = 1,000 and t = 5,000.
5.2. Directed Barabási-Albert Model
Consider a variation of the Barabási-Albert model, where at each time
step a new node arrives and connects with a directed link to a node chosen
with probability
.
Here kin
i
indicates the in-degree of node i and A is the same constant for
all nodes. Each new node has m directed links.
(a) Calculate, using the rate equation approach, the in- and out-de-
gree distribution of the resulting network.
(b) By using the properties of the Gamma and Beta functions, can
you find a power-law scaling for the in-degree distribution?
(c) For A = 0 the scaling exponent of the in-degree distribution is dif-
ferent from γ = 3, the exponent of the Barabási-Albert model. Why?
∏ =
+
∑ +
k
k A
k A
( )
( )
i
i
j j
in
in
in

34
THE BARABÁSI-ALBERT MODEL HOMEWORK
5.3. Copying Model
Use the rate equation approach to show that the directed copying mod-
el leads to a scale-free network with incoming degree exponent γ =
−
−
p
p
2
1
in .
5.4. Growth Without Preferential Attachment
Derive the degree distribution (5.18) of Model A, when a network grows
by new nodes connecting randomly to m previously existing nodes. With
the help of a computer, generate a network of 104
nodes using Model A.
Measure the degree distribution and check that it is consistent with the
prediction (5.18).

ADVANCED TOPICS 5.A
DERIVING THE DEGREE
DISTRIBUTION
SECTION 5.12
A number of analytical techniques are available to calculate the exact
form of the degree exponent (5.11). Next we derive it using the rate equa-
tion approach [12, 13]. The method is sufficiently general to help explore
the properties of a wide range of growing networks. Consequently, the cal-
culations described here are of direct relevance for many systems, from
models pertaining to the WWW [16, 17, 18] to describing the evolution of
the protein interaction network via gene duplication [19, 20].
Let us denote with N(k,t) the number of nodes with degree k at time t.
The degree distribution pk
(t) relates to this quantity via pk
(t) = N(k,t)/N(t).
Since at each time-step we add a new node to the network, we have N = t.
That is, at any moment the total number of nodes equals the number of
timesteps (BOX 5.2).
We write preferential attachment as
where the 2m term captures the fact that in an undirected network each
link contributes to the degree of two nodes. Our goal is to calculate the
changes in the number of nodes with degree k after a new node is added
to the network. For this we inspect the two events that alter N(k,t) and pk
(t)
following the arrival of a new node:
( A new node can link to a degree-k node, turning it into a degree (k+1)
node, hence decreasing N(k,t).
( A new node can link to a degree (k-1) node, turning it into a degree k
node, hence increasing N(k,t).
The number of links that are expected to connect to degree k nodes after
the arrival of a new node is
In (5.32) the first term on the l.h.s. captures the probability that the new
(5.31)
(5.32)
k
k
k
k
mt
( )
2
j
j
∑
Π = =
k
mt
Np t m
k
p t
2
( )
2
( ),
k k
× × =
,
.

36
node will link to a degree-k node (preferential attachment); the second
term provides the total number of nodes with degree k, as the more nodes
are in this category, the higher the chance that a new node will attach to
one of them; the third term is the degree of the incoming node, as the high-
er is m, the higher is the chance that the new node will link to a degree-k
node. We next apply (5.32) to cases (i) and (ii) above:
(i’) The number of degree k nodes that acquire a new link and turn into
(k+1) degree nodes is
(ii’) The number of degree (k-1) nodes that acquire a new link, increasing
their degree to k is
Combining (5.33) and (5.34) we obtain the expected number of degree-k
nodes after the addition of a new node
This equation applies to all nodes with degree km. As we lack nodes
with degree k=0,1, ... , m-1 in the network (each new node arrives with de-
gree m) we need a separate equation for degree-m modes. Following the
same arguments we used to derive (5.35), we obtain
Equations (5.35) and (5.36) are the starting point of the recursive process
that provides pk
. Let us use the fact that we are looking for a stationary
degree distribution, an expectation supported by numerical simulations
(Figure 5.6). This means that in the N = t → ∞ limit, pk
(∞)= pk
. Using this we
can write the l.h.s. of (5.35) and (5.36) as
Therefore the rate equations (5.35) and (5.36) take the form:
Note that (5.37) can be rewritten as
via a k→k+1 variable change.
We use a recursive approach to obtain the degree distribution. That is, we
write the degree distribution for the smallest degree, k=m, using (5.38) and
ADVANCED TOPICS 5.A: DERIVING
THE DEGREE DISTRIBUTION
(5.33)
(5.34)
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
k
p t
2
( )
k
−
−
k
p t
1
2
( ).
k 1
+ + = +
−
−
−
N p t Np t
k
p t
k
p t
( 1) ( 1) ( )
1
2
( )
2
( ).
k k k k
1
+ + = + −
N p t Np t
m
p t
( 1) ( 1) ( ) 1
2
( )
m m m
=
−
+

−
p
k
k
p k m
1
2
k k 1
=
+
p
m
2
2
m
=
+
+
p
k
k
p
3
k k
1
.
.
.
,
(N +1)pk (t +1) Npk (t) Npk ( )+ pk ( ) Npk ( ) = pk ( ) = pk
(N +1)pm(t +1) Npm (t) pm.
,

37
then use (5.39) to calculate pk
for the higher degrees:
At this point we notice a simple recursive pattern: By replacing in the
denomerator m+3 with k we obtain the probability to observe a node with
degree k
which represents the exact form of the degree distribution of the Barabá-
si-Albert model.
Note that:
• For large k (5.41) becomes pk
~ k-3
, in agreement with the numerical re-
sult.
• The prefactor of (5.11) or (5.41) is different from the prefactor of (5.9).
•This form was derived independently in [12] and [13], and the exact
mathematical proof of its validity is provided in [10].
Finally, the rate equation formalism offers an elegant continuum equa-
tion satisfied by the degree distribution [16]. Starting from the equation
we can write
obtaining
One can check that the solution of (5.45) is
ADVANCED TOPICS 5.A: DERIVING
THE DEGREE DISTRIBUTION
(5.41)
(5.42)
(5.43)
(5.44)
(5.45)
(5.46)
=
+
+ +
p
m m
k k k
2 ( 1)
( 1)( 2)
,
k
2 2
k k
=
−
−
− k
p
k
p
k
p
1
1
= − − = − − −
− − −
p k p kp k p k p p
2 ( 1) ( ) [ ],
k k k k k
1 1 1
= − −
−
− −
≈ − −
∂
∂
−
−
−
p p k
p p
k k
p k
p
k
2
( 1)
k k
k k
k
k
1
1
1
= −
∂
∂
p
kp
k
1
2
[ ]
k
k
 −
p k .
k
3
=
+
=
+ +
=
+
+
=
+
+ + +
=
+
+
=
+
+ + +
+
+ +
+ +
p
m
m
p
m
m m
p
m
m
p
m m
m m m
p
m
m
p
m m
m m m
3
2
( 2)( 3)
1
4
2 ( 1)
( 2)( 3)( 4)
2
5
2 ( 1)
( 3)( 4)( 5)
...
m m
m m
m m
1
2 1
3 2
(5.40)
.

ADVANCED TOPICS 5.B
NONLINEAR PREFERENTIAL
ATTACHMENT
SECTION 5.13
In this section we derive the degree distribution of the nonlinear
Barabási-Albert model, governed by the preferential attachment (5.22). We
follow Ref. [13], but we adjust the calculation to cover m 1.
Strictly speaking a stationary degree distribution only exists if α ≤ 1 in
(5.22). For α 1 a few nodes attract a finite fraction of links, as explained in
SECTION 5.7, and we do not have a time-independent pk
. Therefore we limit
ourself to the α ≤ 1 case.
We start with the nonlinear Barabási-Albert model, in which at each
time step a new node is added with m new links. We connect each new link
to an existing node with probability
where ki
is the degree of node i, 0 α ≤ 1 and
is the normalization factor and t=N(t) represents the number of nodes.
Note that and
is the average degree. Since 0 α ≤ 1,
Therefore in the long time limit
constant
whose precise value will be calculated later. For simplicity, we adopt
the notation
Following the rate equation approach introduced in ADVANCED TOPICS
(5.47)
(5.48)
(5.49)
(5.50)
Π ki )=
ki
α
α,t
( )
Μ ( ,t ,t)
t
) =
k
k pk (t)=tμ(
µ 0,t
( )= ∑
k
pk (t) =1 µ 1,t
( )= ∑
k
kpk (t) = k〉 = 2mt / N
µ 0,t
( )≤ µ α,t
( )≤ µ 1,t
( )
µ(α,t →∞) =
µ ≡ µ(α,t →∞)
,
.
.

39
5.A, we write the rate equation for the network’s degree distribution as
The first term on the r.h.s. describes the rate at which nodes with degree
(k-1) gain new links; the second term describes the loss of degree-k nodes
when they gain new links, turning into (k+1) degree nodes; the last term
represents the newly added nodes with degree m.
Asymptotically, in the t→∞ limit, we can write pk
=pk
(t + 1)=pk
(t). Substi-
tuting k=m in (5.51) we obtain:
For k m
Solving (5.53) recursively we obtain
To determine the large k behavior of pk
we take the logarithm of (5.57):
Using the series expansion we obtain
We approximate the sum over j with the integral
which in the special case of nα =1 becomes
ADVANCED TOPICS 5.B
NONLINEAR PREFERENTIAL ATTACHMENT
(5.51)
(5.52)
(5.53)
(5.54)
(5.55)
(5.57)
(5.56)
(5.58)
(5.59)
(5.60)
(5.61)
pk (t t
+1)
(t +1) =
m
µ( ,t)
[(k 1) pk 1(t) k pk (t)
pk (t)+ ]+ k,m.
ln pk = ln(µ / m)−α lnk − ∑
j=m
k
∑
n=1
∞
(−1)n+1
n
(µ / m)n
j−nα
j=m
k
∑j−1
≈
m
k
∫x−1
dx = lnk −lnm.
pk =
(k −1)α
µ / m+ kα
pk−1 .
pm+1 =
m
µ / m+(m+1)
µ / m
µ / m+ m
,
pm =
µ / m
µ / m+ mα
,
.
pk =
µ / m
k
k
j=m
1+
µ / m
j
1
ln 1+ x
( )=
n=1
1
( )
n+1
/ n xn
k
j=m
jX
n
k
m
x n
dx =
1
1 n
(k1 n
m1 n
)
pm = −
m
µ
− mα
pm +1
pm = −
µ / m
µ / m+ mα
.
,
pk =
m
µ
[(k −1)α
pk−1 − kα
pk ] ,
ln pk = ln(µ / m) lnk
k
j=m
1+
µ / m
j .
.

40
Hence we obtain
Consequently the degree distribution has the form
where
The vanishing terms in the exponential do not influence the k → ∞ asymp-
totic behavior, being relevant only if 1−nα ≥ 1. Consequently pk
depends on
α as:
That is, for 1/2 α 1 the degree distribution follows a stretched exponen-
tial. As we lower α, new corrections start contributing each time α becomes
smaller than 1/n, where n is an integer.
For α→1 the degree distribution scales as k−3
, as expected for the Barabá-
si-Albert model. Indeed for α = 1 we have μ=2, and
Therefore pk
~ k−1
exp(−2lnk) = k−3
.
Finally we calculate . For this we write the sum (5.58)
We obtain μ(α) by solving (5.68) numerically.
ADVANCED TOPICS 5.B
NONLINEAR PREFERENTIAL ATTACHMENT
(5.62)
(5.63)
(5.64)
(5.65)
(5.66)
(5.67)
(5.68)
ln pk = ln(µ / m) lnk
n=1
( 1)n+1
n
(µ / m)n
1 n
(k1 n
m1 n
)
pk = α k−α
e
−∑
∞
n=1
(−1)n+1
n
(µ/m)n
1−nα
k1−nα
,
µ α
( )= ∑
j
jα
pj
lim
k
α
k
1
ln .
α
α
1
1
−
=
→
−
pk
k e
µ/m
1
k1
1/ 2 1
k
1
2
+
1
2
(
µ
m
)2
e
1
2
µ
m
k 2
=1/ 2
k e
µ/m
1
k1
+
1
2
(µ/m)2
1 2
k1 2
1/ 3 1/ 2
.
α =
µ
m
e
∑
∞
n=1
(−1)n+1
n
(µ/m)n
1−nα
m1−nα
.
1=
1
m
∑
k=m
∞
∏
j=m
k
1+
µ(α) / m
jα






−1
.
∑
k=m
∞
kα
pk = ∑
k=m
∞
µ(α)
m
∏
j=m
k
1+
µ(α) / m
jα
⎛
⎝
⎜
⎞
⎠
⎟
−1
,

ADVANCED TOPICS 5.C
THE CLUSTERING COEFFICIENT
SECTION 5.14
In this section we derive the average clustering coefficient, (5.30), for
the Barabási-Albert model. The derivation follows an argument proposed
by Klemm and Eguiluz [35], supported by the exact calculation of Bollobás
[36].
We aim to calculate the number of triangles expected in the model,
which can be linked to the clustering coefficient (SECTION 2.10). We denote
the probability to have a link between node i and j with P(i,j). Therefore,
the probability that three nodes i, j, l form a triangle is P(i,j)P(i,l)P(j,l). The
expected number of triangles in which node l with degree kl
participates
is thus given by the sum of the probabilities that node l participates in tri-
angles with arbitrary chosen nodes i and j in the network. We can use the
continuous degree approximation to write
To proceed we need to calculate P(i,j), which requires us to consider how
the Barabási-Albert model evolves. Let us denote the time when node j ar-
rived with tj
=j, which we can do as in each time step we added only one new
node (event time, BOX 5.2). Hence the probability that at its arrival node j
links to node i with degree ki
is given by preferential attachment
Using (5.7), we can write
where we used the fact that the arrival time of node j is tj
=j and the arrival
time of node i is ti
= i. Hence (5.70) now becomes
(5.69)
(5.70)
(5.71)
Nr di djP i j P i l P j l
( ) ( , ) ( , ) ( , )
l
i
N
j
N
1 1
∫ ∫
=
= =

( )
= =
P i j m k j m
k j
k (j)
m
k j
mj
( , ) ( ( ))
( )
2
i
i
l
l
j
i
1
∑
= Π
=
ki (t) = m
t
ti






1
2
= m
j
i






1
2
,
.
.

42
Using this result we calculate the number of triangles in (5.69), writing
The clustering coefficient can be written as , hence we obtain
To simplify (5.74), we note that according to (5.7) we have
which is the degree of node l at time t = N. Hence, for large kl
we have
allowing us to write the clustering coefficient of the Barabási-Albert model
as
which is independent of l, therefore we obtain the result (5.30).
ADVANCED TOPICS 5.C
THE CLUSTERING COEFFICIENT
(5.73)
(5.74)
(5.75)
(5.76)
(5.77)
Nr di djP i j P i l P j l
( ) ( , ) ( , ) ( , )
l
i
N
j
N
1 1
∫ ∫
=
= =

m
di dj ij il jl
8
( ) ( ) ( )
i
N
j
N
3
1 1
1
2
1
2
1
2
∫ ∫
=
= =
− − −
m
l
di
i
dj
j
m
l
N
8 8
(ln )
i
N
j
N
3
1 1
3
2
∫ ∫
= =
= =
C
Nr
k k
2 ( )
( 1)
l
l
l l

=
−
(N)−
Cl
m
l
N
k (N) k
4
(ln )
( 1)
l l
3
2
=
kl (N) = m
N
⎛
⎝
⎜
⎞
⎠
⎟
1
2
( 1 k

l
l
2 2
≈ =
()−
k () k )
l l (
Cl
=
m
4
(ln )

2
,
,
,
.
P i j
m
ij
( , )
2
( ) .
1
2
=
−
(5.72)

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BIBLIOGRAPHY

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